Double total domination number of Cartesian product of paths

A vertex set $ S $ of a graph $ G $ is called a double total dominating set if every vertex in $ G $ has at least two adjacent vertices in $ S $. The double total domination number $ \gamma_{\times 2, t}(G) $ of $ G $ is the minimum cardinality over all the double total dominating sets in $ G $. Let $ G \square H $ denote the Cartesian product of graphs $ G $ and $ H $. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i = 2, 3 $, and give lower and upper bounds of $ \gamma_{\times 2, t}(P_i\square P_n) $ for $ i \geq 4 $.